Central limit theorems are proved for martingales and near-martingales without the existence of moments or the full Lindeberg condition. These theorems are extended to invariance principles with a discussion of both random and nonrandom norming.
"Dependent Central Limit Theorems and Invariance Principles." Ann. Probab. 2 (4) 620 - 628, August, 1974. https://doi.org/10.1214/aop/1176996608